why is the area of the intersection of the cylinder and the plane Pi?

Question
I have a cylinder equation (x-1+az)^2+(y+bz)^2=1. The intersection of
the cylinder and the YOZ plane should be bigger than the base when it
is an ellipse. The base is the circle (x-1)^2+y^2=1 with area Pi. The
intersection is (az-1)^2+(y+bz)^2=1. I tried different a's and b's,
The area is always Pi, for example letting a=1 and b=10. The
calculation is done by using Wolfram|Alpha: Computational Knowledge Engine. The short axis of
this ellipse seems smaller than 2. I think since the
base circle has radius 1, the ellipse should have short axis 2.

The result is very counter intuitive. When one cuts the cylinder
with a plane and the result is an ellipse, the area of the ellipse
should be bigger than the base circle.